Factors, Multiples and Primes - Short Problems

A collection of short Stage 3 and 4 problems on factors, multiples and primes



Four or Five

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 19 - 2012
The diagram shows a large rectangle composed of 9 smaller rectangles. If each of these rectangles has integer sides, what could the area of the large rectangle be?

Seven from Nine

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 21 - 2013
In how many ways can seven of the numbers 1-9 be chosen such that they add up to a multiple of 3?

Factor Trio

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 45 - 2013
Which of the numbers shown is the product of exactly 3 distinct prime factors?

Back of the Queue

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 48 - 2013
What is the remainder when the number 743589×301647 is divided by 5?

Calculation 2000

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 49 - 2013
What is the value of 2000 + 1999 × 2000?

Prime Order

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 24 - 2006
How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?

Essential Supplies

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 25 - 2006
Chocolate bars come in boxes of 5 or boxes of 12. How many boxes do you need to have exactly 2005 chocolate bars?

Powerful Zeros

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 32 - 2006
How many zeros are there at the end of $3^4 \times 4^5 \times 5^6$?

Cakes and Buns

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 41 - 2006
Helen buys some cakes and some buns for her party. Can you work out how many of each she buys?

Find from Factors

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 35 - 2006
A number has exactly eight factors, two of which are 21 and 35. What is the number?

Missing Digit

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 37 - 2006
What digit must replace the star to make the number a nultiple of 11?

Coin Collection

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 5 - 2007,br />hen coins are put into piles of six 3 remain and in piles of eight 7 remain. How many remain when they are put into piles of 24?

Nine Sum to Square

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 32 - 2007
One of these numbers is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?

Producing Zeros

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 10 - 2008
If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?

Punky Fish

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 12 - 2008
A male punky fish has 9 stripes and a female punky fish has 8 stripes. I count 86 stripes on the fish in my tank. What is the ratio of male fish to female fish?

Expanding Zeros

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 22 - 2008
The following sequence continues indefinitely... Which of these integers is a multiple of 81?

Leftovers

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 26 - 2008
If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?

Prime Pairs

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 41 - 2008
How many pairs of numbers of the form $x$, $2x+1$ are there in which both numbers are prime numbers less than $100$?

School Dinners

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 1 - 2009
Our school dinners offer the same choice each day, and each day I try a new option. How long will it be before I eat the same meal again?

Little Goldbach

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 11 - 2009
How many of the numbers 1 to 20 are not the sum of two primes?

Trailing Zeros

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 13 - 2009
How many zeros does 50! have at the end?

Cinema Costs

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 41 - 2009
At a cinema a child's ticket costs £$4.20$ and an adult's ticket costs £$7.70$. How much did is cost this group of adults and children to see a film?

Flora the Florist

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 45 - 2009
Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?

Magician's Hat

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 52 - 2009
Gar the Magician plays a card trick on his friends Kan and Roo. Can you work out his trick and find out the sum on Kan's cards?

Three Primes

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 6 - 2010
Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?

Square LCM

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 7 - 2010
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

Factorised Factorial

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 17 - 2010
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?

17s and 23s

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 22 - 2010
Can you form this 2010-digit number...

HCF Expression

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 30 - 2010
Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.

Product 100

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 38 - 2010
The product of four different positive integers is 100. What is the sum of these four integers?

Factor List

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 48 - 2010
Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?

Adjacent Factors

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 5 - 2011
Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

Sticky Fingers

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 7 - 2011
Roo wants to puts stickers on the cuboid he has made from little cubes. Will he have any stickers left over?

Square Product

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 10 - 2011
Will this product give a perfect square?

Indivisible

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 25 - 2011
Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?

Many Clues, One Answer

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 35 - 2011
You are given lots of clues about a number. Can you work out what it is?

Divisible Palindrome

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 5 - 2014
What is the sum of the digits of the largest 4-digit palindromic number which is divisible by 15?

Digital Division

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 15 - 2014
How many three digit numbers formed with three different digits from 0, 1, 2, 3 and 5 are divisible by 6?

Almost a Million

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 26 - 2014
Which of the given numbers are divisible by 6?

Supercomputer

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 28 - 2014
What is the units digit of the given expression?

One Less Remainder

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 39 - 2014
A whole number less than 100 gives remainders of 2, 3 and 4 when divided by 3, 4 and 5. What is the remainder when it is divided by 7?

Primes and Six

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 1 - 2015
If $p$ and $q$ are prime numbers greater than $3$ and $q=p+2$, prove that $pq+1$ is divisible by $36$.

Factor Sum

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 4 - 2015
Given any positive integer n, Paul adds together the factors of n, apart from n itself. Which of the numbers 1, 3, 5, 7 and 9 can never be Paul's answer?

Tricky Customer

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 6 - 2015
Charlie doesn't want his new house number to be divisible by 3 or 5. How many choices of house does he have?